https://file.aiquickdraw.com/mathbot/file/1726184642197hy3qpst2.jpg

Here are the functions and operations as shown in the image:

Given functions:
- \( f(x) = 6x^2 - 35x + 49 \)
- \( g(x) = \frac{4x}{3} + 2 \)
- \( j(x) = \frac{2}{x} - 1 \)
- \( k(x) = 3x - 7 \)
- \( m(x) = 4x + 22 \)
- \( h(x) = x^2 + 3x - 4 \)

Tasks:
1. \( (h - k)(x) \)
2. \( \left(\frac{g}{j}\right)(x) \)
3. \( (f \cdot k)(x) \)
4. \( (k + m)(x) \)
5. \( h(k) - (k \cdot m)(x) \)

### Let's solve them step by step:

#### 1. \( (h - k)(x) \)
This represents the difference between the functions \( h(x) \) and \( k(x) \):
\[
(h - k)(x) = h(x) - k(x) = (x^2 + 3x - 4) - (3x - 7)
\]
Simplify the expression:
\[
= x^2 + 3x - 4 - 3x + 7 = x^2 + 3x - 3x - 4 + 7 = x^2 + 3
\]
So, \( (h - k)(x) = x^2 + 3 \).

#### 2. \( \left(\frac{g}{j}\right)(x) \)
This represents the division of the functions \( g(x) \) and \( j(x) \):
\[
\left(\frac{g}{j}\right)(x) = \frac{g(x)}{j(x)} = \frac{\frac{4x}{3} + 2}{\frac{2}{x} - 1}
\]
To simplify this, multiply the numerator and denominator by \( x \):
\[
\frac{\left(\frac{4x}{3} + 2\right) \cdot x}{\left(\frac{2}{x} - 1\right) \cdot x} = \frac{\frac{4x^2}{3} + 2x}{2 - x}
\]
So, \( \left(\frac{g}{j}\right)(x) = \frac{\frac{4x^2}{3} + 2x}{2 - x} \).

#### 3. \( (f \cdot k)(x) \)
This represents the product of the functions \( f(x) \) and \( k(x) \):
\[
(f \cdot k)(x) = f(x) \cdot k(x) = (6x^2 - 35x + 49)(3x - 7)
\]
Expanding this:
\[
(6x^2 - 35x + 49)(3x - 7) = 6x^2(3x) + 6x^2(-7) - 35x(3x) - 35x(-7) + 49(3x) + 49(-7)
\]
\[
= 18x^3 - 42x^2 - 105x^2 + 245x + 147x - 343
\]
Combine like terms:
\[
= 18x^3 - 147x^2 + 392x - 343
\]
So, \( (f \cdot k)(x) = 18x^3 - 147x^2 + 392x - 343 \).

#### 4. \( (k + m)(x) \)
This represents the sum of the functions \( k(x) \) and \( m(x) \):
\[
(k + m)(x) = k(x) + m(x) = (3x - 7) + (4x + 22)
\]
Simplify the expression:
\[
= 3x - 7 + 4x + 22 = 7x + 15
\]
So, \( (k + m)(x) = 7x + 15 \).

#### 5. \( h(k) - (k \cdot m)(x) \)
This is a two-part problem:
1. First, evaluate \( h(k(x)) \), which means substituting \( k(x) = 3x - 7 \) into \( h(x) \):
\[
h(k(x)) = h(3x - 7) = (3x - 7)^2 + 3(3x - 7) - 4
\]
Expanding the square and simplifying:
\[
= (9x^2 - 42x + 49) + (9x - 21) - 4 = 9x^2 - 42x + 49 + 

Given the functions:
1. f(x) = 6x^2 - 35x + 49
2. g(x) = (4x/3) + 2
3. j(x) = 2/x - 1
4. k(x) = 3x - 7
5. m(x) = 4x + 22
6. h(x) = x^2 + 3x - 4

Perform the following operations:
1. (h - k)(x)
2. (g/j)(x)
3. (f * k)(x)
4. (k + m)(x)
5. h(k) - (k * m)(x)

Learn how to perform operations with functions such as addition, subtraction, multiplication, and division. This problem set includes a variety of functions and demonstrates step-by-step solutions for operations like (h - k)(x), (g/j)(x), (f * k)(x), and more. Suitable for students in Grades 9-12.

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